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〖<font color="#0033CC">lemma7.1</font>〗Let Tavg(n) be the expected time for quicksort to sort
a file with n records. Then there exists a constant k such that
<img src="image/text7-h.gif" width="130" height="23" align="middle"> for n >= 2.
<img src="image/text7-i.gif" width="50" height="23" align="middle"> <img src="image/less.gif" width="10" height="13"> cn + <img src="image/text7-j.gif" width="205" height="39" align="middle"> = cn + <img src="image/text7-k.gif" width="91" height="39" align="middle">
<font color="#FF0000">proof:</font>
<font color="#0033CC">induction base</font>:
for n = 2, <img src="image/text7-l.gif" width="197" height="23">
induction hypothesis:
<img src="image/text7-h.gif" width="130" height="23" align="middle"> for 1 <img src="image/less.gif" width="10" height="13"> n < m
<font color="#0033CC">induction step</font>:
<img src="image/text7-m.gif" width="410" height="43" align="middle">
<img src="image/text7-o.gif" width="460" height="45" align="middle">
<img src="image/text7-n.gif" width="295" height="42" align="middle"> for m >= 2
<font color="#FF0000">stack space</font>: <font color="#FF0000">O(log n)</font>
<font color="#FF0000">variation</font>
quicksort using a median of three:
povit = median {<img src="image/text7-k0.gif" width="221" height="28" align="middle"> }</font></b></pre>
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